桶排序

桶排序
概況
類別	排序算法
資料結構	數組
複雜度
平均時間複雜度	$O(n+k)$，$k$為桶數
最壞時間複雜度	$O(n^{2})$
空間複雜度	$O(n\cdot k)$
最佳解	$O(n)$
相關變量的定義

元素分配到桶中

對桶中元素排序

桶排序（英語：Bucket sort）或所謂的箱排序，是一個排序演算法，工作的原理是將陣列分到有限數量的桶裡。每個桶再個別排序（有可能再使用別的排序演算法或是以遞迴方式繼續使用桶排序進行排序）。桶排序是鴿巢排序的一種歸納結果。當要被排序的陣列內的數值是均勻分配的時候，桶排序使用線性時間（$\Theta (n)$）。

桶排序以下列程序進行：

1. 設置一個定量的陣列當作空桶子。
2. 尋訪序列，並且把項目一個一個放到對應的桶子去。
3. 對每個不是空的桶子進行排序。
4. 從不是空的桶子裡把項目再放回原來的序列中。

偽代碼

function bucket-sort(array, n) is
  buckets ← new array of n empty lists
  for i = 0 to (length(array)-1) do
    insert array[i] into buckets[msbits(array[i], k)]
  for i = 0 to n - 1 do
    next-sort(buckets[i])
  return the concatenation of buckets[0], ..., buckets[n-1]

C++實現算法

假設數據分布在[0，100)之間，每個桶內部用鍊表表示，在數據入桶的同時插入排序。然後把各個桶中的數據合併。

#include<iterator>
#include<iostream>
#include<vector>
using namespace std;
const int BUCKET_NUM = 10;

struct ListNode{
	explicit ListNode(int i=0):mData(i),mNext(NULL){}
	ListNode* mNext;
	int mData;
};

ListNode* insert(ListNode* head,int val){
	ListNode dummyNode;
	ListNode *newNode = new ListNode(val);
	ListNode *pre,*curr;
	dummyNode.mNext = head;
	pre = &dummyNode;
	curr = head;
	while(NULL!=curr && curr->mData<=val){
		pre = curr;
		curr = curr->mNext;
	}
	newNode->mNext = curr;
	pre->mNext = newNode;
	return dummyNode.mNext;
}


ListNode* Merge(ListNode *head1,ListNode *head2){
	ListNode dummyNode;
	ListNode *dummy = &dummyNode;
	while(NULL!=head1 && NULL!=head2){
		if(head1->mData <= head2->mData){
			dummy->mNext = head1;
			head1 = head1->mNext;
		}else{
			dummy->mNext = head2;
			head2 = head2->mNext;
		}
		dummy = dummy->mNext;
	}
	if(NULL!=head1) dummy->mNext = head1;
	if(NULL!=head2) dummy->mNext = head2;
	
	return dummyNode.mNext;
}

void BucketSort(int n,int arr[]){
	vector<ListNode*> buckets(BUCKET_NUM,(ListNode*)(0));
	for(int i=0;i<n;++i){
		int index = arr[i]/BUCKET_NUM;
		ListNode *head = buckets.at(index);
		buckets.at(index) = insert(head,arr[i]);
	}
	ListNode *head = buckets.at(0);
	for(int i=1;i<BUCKET_NUM;++i){
		head = Merge(head,buckets.at(i));
	}
	for(int i=0;i<n;++i){
		arr[i] = head->mData;
		head = head->mNext;
	}
}

Java實現算法

	private int indexFor(int a, int min, int step) {
		return (a - min) / step;
	}

	public void bucketSort(int[] arr) {

		int max = arr[0], min = arr[0];
		for (int a : arr) {
			if (max < a)
				max = a;
			if (min > a)
				min = a;
		}
		// 該值也可根據實際情況選擇
		int bucketNum = max / 10 - min / 10 + 1;
		List buckList = new ArrayList<List<Integer>>();
		// create bucket
		for (int i = 1; i <= bucketNum; i++) {
			buckList.add(new ArrayList<Integer>());
		}
		// push into the bucket
		for (int i = 0; i < arr.length; i++) {
			int index = indexFor(arr[i], min, 10);
			((ArrayList<Integer>) buckList.get(index)).add(arr[i]);
		}
		ArrayList<Integer> bucket = null;
		int index = 0;
		for (int i = 0; i < bucketNum; i++) {
			bucket = (ArrayList<Integer>) buckList.get(i);
			insertSort(bucket);
			for (int k : bucket) {
				arr[index++] = k;
			}
		}

	}

	// 把桶內元素插入排序
	private void insertSort(List<Integer> bucket) {
		for (int i = 1; i < bucket.size(); i++) {
			int temp = bucket.get(i);
			int j = i - 1;
			for (; j >= 0 && bucket.get(j) > temp; j--) {
				bucket.set(j + 1, bucket.get(j));
			}
			bucket.set(j + 1, temp);
		}
	}

JavaScript實現算法

Array.prototype.bucketSort = function(num) {
  function swap(arr, i, j) {
    const temp = arr[i]
    arr[i] = arr[j]
    arr[j] = temp
  }
  const max = Math.max(...this)
  const min = Math.min(...this)
  const buckets = []
  const bucketsSize = Math.floor((max - min) / num) + 1
  for(let i = 0; i < this.length; i++) {
    const index = ~~(this[i] / bucketsSize)
    !buckets[index] && (buckets[index] = [])
    buckets[index].push(this[i])
    let l = buckets[index].length
    while(l > 0) {
      buckets[index][l] < buckets[index][l - 1] && swap(buckets[index], l, l - 1)
      l--
    }
  }
  let wrapBuckets = []
  for(let i = 0; i < buckets.length; i++) {
    buckets[i] && (wrapBuckets = wrapBuckets.concat(buckets[i]))
  }
  return wrapBuckets
}
const arr = [11, 9, 6, 8, 1, 3, 5, 1, 1, 0, 100]
console.log(arr.bucketSort(10))

JavaScript實現遞歸算法

Array.prototype.bucketSort = function()
{
  var start = 0;
  var size = this.length;
  var min = this[0];
  var max = this[0]; 
  for (var i = 1; i < size; i++){
    if (this[i] < min){min = this[i];}
    else{ if(this[i] > max){max = this[i];} }
  }
  if (min != max){
    var bucket = new Array(size);
    for (var i = 0; i < size; i++){bucket[i] = new Array();}
    var interpolation = 0;
    for (var i = 0; i < size; i++){
      interpolation = Math.floor(((this[i] - min) / (max - min)) * (size - 1));
      bucket[interpolation].push(this[i]);
    }
    for (var i = 0; i < size; i++){
      if (bucket[i].length > 1){bucket[i].bucketSort();}//遞歸
      for(var j = 0; j < bucket[i].length; j++){this[start++] = bucket[i][j];}
    }
  }
};

Python 3.10 實現演算法

def is_sort(arr: list) -> bool:
    for i in range(len(arr) - 1):
        if arr[i] > arr[i + 1]:
            return False
    return True

def bucket_sort(arr: list, is_sub_bucket: bool = False):
    if is_sort(arr):
        return

    bucket_num: int = max(arr) // 10 + 1 if not is_sub_bucket else 10
    buckets: list[list] = [[] for _ in range(bucket_num)]

    for a in arr:
        i: int = a // 10 if not is_sub_bucket else a % 10
        buckets[i].append(a)
    
    arr.clear()
    for bucket in buckets:
        bucket_sort(bucket, is_sub_bucket=True)
        arr += bucket

if __name__ == '__main__':
    arr = [29, 25, 3, 49, 9, 37, 21, 43]
    bucket_sort(arr)
    print(arr)

外部連結

• Bucket Sort in C++
• Bucket Sort & PHP Sample

參考

• Thomas H. Cormen，Charles E. Leiserson，Ronald L. Rivest，and Clifford Stein。Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 8.4: Bucket sort, pp.174–177.
• Paul E. Black "Postman's Sort" （頁面存檔備份，存於網際網路檔案館） from Dictionary of Algorithms and Data Structures at NIST。
• Robert Ramey The Postman's Sort （頁面存檔備份，存於網際網路檔案館） C Users Journal Aug. 1992